arXiv:1309.0154v1 [math.FA] 31 Aug 2013
Some New Paranormed Difference Sequence Spaces
Derived by Fibonacci Numbers
Emrah Evren Karaa,∗, Serkan Demirizb
a
Department of Mathematics, Duzce University, 81620, Duzce, Turkey
b
Department of Mathematics, Gaziosmanpa¸sa University, 60250 Tokat, Turkey
Abstract
In this study, we define new paranormed sequence spaces by the sequences of Fibonacci numbers. Furthermore, we compute the α−, β− and γ− duals and obtain bases for these sequence spaces. Besides this, we characterize the matrix transformations from the new paranormed sequence spaces to the Maddox’s spaces c0(q), c(q), ℓ(q) and ℓ∞(q).
Keywords: Paranormed sequence spaces, Matrix transformations, The
sequences of Fibonacci numbers
1. Preliminaries,background and notation
By ω, we shall denote the space of all real valued sequences. Any vector subspace of ω is called as a sequence space. We shall write ℓ∞, c and c0 for the
spaces of all bounded, convergent and null sequences, respectively. Also by bs, cs, ℓ1 and ℓp ; we denote the spaces of all bounded, convergent, absolutely
and p− absolutely convergent series, respectively; 1 < p < ∞.
A linear topological space X over the real field R is said to be a para-normed space if there is a subadditive function g : X → R such that g(θ) = 0, g(x) = g(−x) and scalar multiplication is continuous,i.e., |αn−α| → 0 and
∗Corresponding Author (Tel: +90 356 252 16 16, Fax: +90 356 252 15 85)
Email addresses: eevrenkara@hotmail.com(Emrah Evren Kara), serkandemiriz@gmail.com(Serkan Demiriz)
g(xn− x) → 0 imply g(αnxn − αx) → 0 for all α′s in R and all x’s in X,
where θ is the zero vector in the linear space X.
Assume here and after that (pk) be a bounded sequences of strictly
pos-itive real numbers with sup pk = H and M = max{1, H}. Then, the linear
spaces c(p), c0(p), ℓ∞(p) and ℓ(p) were defined by Maddox [1, 2] (see also
Simons [3] and Nakano [4]) as follows:
c(p) = nx = (xk) ∈ ω : lim k→∞|xk− l| pk = 0 for some l ∈ Co, c0(p) = n x = (xk) ∈ ω : lim k→∞|xk| pk = 0o, ℓ∞(p) = x = (xk) ∈ ω : sup k∈N|xk| pk < ∞ and ℓ(p) =nx = (xk) ∈ ω : X k |xk|pk < ∞ o , which are the complete spaces paranormed by
h1(x) = sup k∈N|xk| pk/M iff inf pk > 0 and h2(x) = X k |xk|pk 1/M ,
respectively. We shall assume throughout that p−1 k + (p
′
k)
−1 = 1 provided
1 < inf pk < H < ∞. For simplicity in notation, here and in what follows,
the summation without limits runs from 0 to ∞. By F and Nk, we shall
denote the collection of all finite subsets of N and the set of all n ∈ N such that n ≥ k, respectively. We write by U for the set of all sequences u = (un)
such that un6= 0 for all n ∈ N. For u ∈ U, let 1/u = (1/un).
For the sequence spaces X and Y , define the set S(X, Y ) by
S(X, Y ) = {z = (zk) : xz = (xkzk) ∈ Y for all x ∈ X}. (1)
With the notation of (1), the α−, β− and γ− duals of a sequence space X, which are respectively denoted by Xα, Xβ and Xγ, are defined by
Xα = S(X, ℓ1), Xβ = S(X, cs) and Xγ = S(X, bs).
Let (X, h) be a paranormed space. A sequence (bk) of the elements of X
sequence (αk) of scalars such that h x − n X k=0 αkbk ! → 0 as n → ∞.
The seriesPαkbk which has the sum x is then called the expansion of x with
respect to (bn) and written as x = Pαkbk. Let X, Y be any two sequence
spaces and A = (ank) be an infinite matrix of real numbers ank,where n, k ∈
N. Then, we say that A defines a matrix mapping from X into Y , and we denote it by writing A : X → Y , if for every sequence x = (xk) ∈ X the
sequence Ax = ((Ax)n), the A-transform of x, is in Y , where
(Ax)n=
X
k
ankxk, (n ∈ N). (2)
By (X : Y ), we denote the class of all matrices A such that A : X → Y . Thus, A ∈ (X : Y ) if and only if the series on the right-hand side of (2) converges for each n ∈ N and every x ∈ X, and we have Ax = {(Ax)n}n∈N∈ Y for all
x ∈ X. A sequence x is said to be A- summable to α if Ax converges to α which is called as the A- limit of x.
For a sequence space X, the matrix domain XA of an infinite matrix A
is defined by
XA= {x = (xk) ∈ ω : Ax ∈ X}.
The approach constructing a new paranormed sequence space by means of the matrix domain of a particular limitation method has recently been employed by Malkowsky [5], Altay and Ba¸sar [6, 7], F. Ba¸sar et al., [8], Aydın and Ba¸sar [9, 10].
Define the sequence {fn}∞n=0 of Fibonacci numbers given by the linear
recurrence relations
f0 = f1 = 1 and fn= fn−1+ fn−2, n ≥ 2.
In modern science and particularly physics, there is quite an interest in the theory and applications of Fibonacci numbers . The ratio of the successive Fibonacci numbers is as known golden ratio. There are many applications of golden ratio in many places of mathematics and physics, in general theory of high energy particle theory [11]. Also, some basic properties of Fibonacci numbers [12] are given as follows:
lim n→∞ fn+1 fn = 1 + √ 5 2 = α (golden ratio)
n X k=0 fk = fn+2− 1 (n ∈ N) and X k 1 fk converges fn−1fn+1− fn2 = (−1)n+1 (n ≥ 1) (Cassini formula).
Substituting for fn+1in Cassini’s formula yields fn−12 +fnfn−1−fn2 = (−1)n+1.
Let fn be the nth Fibonacci number for every n ∈ N. Then, the infinite
Fibonacci matrix bF = ( bfnk) is defined by
b fnk = −fn+1f n (k = n − 1), fn fn+1 (k = n), 0 (0 ≤ k < n − 1 or k > n) where n, k ∈ N [13].
The main purpose of this study is to introduce the sequence spaces c0( bF , p), c( bF , p) ,ℓ∞( bF , p) and ℓ( bF , p) which are the set of all sequences whose
b
F −transforms are in the spaces c0(p), c(p), ℓ∞(p) and ℓ(p), respectively. Also,
we have investigated some topological structures, which have completeness, the α−, β− and γ− duals, and the bases of these sequence spaces. Besides this, we characterize some matrix mappings on these spaces.
2. The Paranormed Fibonacci Difference Sequence Spaces
In this section, we define the new sequence spaces c0( bF , p), c( bF , p) ,ℓ∞( bF , p)
and ℓ( bF , p) by using the sequences of Fibonacci numbers, and prove that these sequence spaces are the complete paranormed linear metric spaces and compute their α−, β− and γ− duals. Moreover, we give the basis for the spaces c0( bF , p), c( bF , p) and ℓ( bF , p) .
For a sequence space X, the matrix domain XA of an infinite matrix A
is defined by
XA= {x = (xk) ∈ ω : Ax ∈ X}. (3)
In [14], Choudhary and Mishra have defined the sequence space ℓ(p) which consists of all sequences such that S-transforms are in ℓ(p), where S = (snk)
is defined by
snk =
1, (0 ≤ k ≤ n), 0, (k > n).
Ba¸sar and Altay [15] have recently examined the space bs(p) which is for-merly defined by Ba¸sar in [16] as the set of all series whose sequences of partial sums are in ℓ∞(p). More recently, Altay and Ba¸sar have studied the sequence
spaces rt(p), rt
∞(p) in [17] and rct(p), r0t(p) in [18] which are derived by the
Riesz means from the sequence spaces ℓ(p), ℓ∞(p), c(p) and c0(p) of Maddox,
respectively. With the notation of (3), the spaces ℓ(p), bs(p), rt(p), rt
∞(p), rct(p) and rt 0(p) may be redefined by ℓ(p) = [ℓ(p)]S, bs(p) = [ℓ∞(p)]S, rt(p) = [ℓ(p)]Rt, r∞t (p) = [ℓ∞(p)]Rt, rt c(p) = [c(p)]Rt, rt 0(p) = [c0(p)]Rt.
Following Choudhary and Mishra [14], Ba¸sar and Altay [15], Altay and Ba¸sar [17, 18], we define the sequence spaces c0( bF , p), c( bF , p) ,ℓ∞( bF , p) and
ℓ( bF , p) by c0( bF , p) = n x = (xk) ∈ ω : lim n→∞ ffn n+1 xn− fn+1 fn xn−1 pn = 0o c( bF , p) = nx = (xk) ∈ ω : ∃l ∈ C ∋ lim n→∞ ffn n+1 xn− fn+1 fn xn−1− l pn = 0o ℓ∞( bF , p) = n x = (xk) ∈ ω : sup n∈N ffn n+1 xn− fn+1 fn xn−1 pn < ∞o and ℓ( bF , p) =nx = (xk) ∈ ω : X n ffn n+1 xn− fn+1 fn xn−1 pn < ∞o.
In the case (pk) = e = (1, 1, 1, ...), the sequence spaces c0( bF , p), c( bF , p)
,ℓ∞( bF , p) and ℓ( bF , p) are , respectively, reduced to the sequence spaces
c0( bF ), c( bF ), ℓ∞( bF ) and ℓp( bF ) which are introduced by E.E.Kara [13] and
M. Ba¸sarır et al. [19].
With the notation (3), we may redefine the spaces c0( bF , p), c( bF , p) ,ℓ∞( bF , p)
and ℓ( bF , p) as follows:
c0( bF , p) = {c0(p)}Fb, c( bF , p) = {c(p)}Fb,
Define the sequence y = (yk), which will be frequently used as the
b
F −transform of a sequence x = (xk), i.e.
yk= bFk(x) = fk fk+1 xk− fk+1 fk xk−1; (k ∈ N0). (4)
Since the proof may also be obtained in the similar way as for the other spaces, to avoid the repetition of the similar statements, we give the proof only for one of those spaces. Now, we may begin with the following theorem which is essential in the study.
Theorem 1. (i) The sequence spaces c0( bF , p), c( bF , p) and ℓ∞( bF , p) are the
complete linear metric spaces paranormed by g, defined by
g(x) = sup k∈N ffk k+1 xk− fk+1 fk xk−1 pk/M .
g is a paranorm for the spaces c( bF , p) and ℓ∞( bF , p) only in the trivial case
inf pk > 0 when c( bF , p) = c( bF ) and ℓ∞( bF , p) = ℓ∞( bF ).
(ii) ℓp( bF ) is a complete linear metric space paranormed by
g∗ (x) = X k ffk k+1 xk− fk+1 fk xk−1 pk1/M .
Proof. We prove the theorem for the space c0( bF , p). The linearity of c0( bF , p) with respect to the coordinatewise addition and scalar multiplication
follows from the following inequalities which are satisfied for x, z ∈ c0( bF , p)
(see [20, p.30]): sup k∈N ffk k+1 (xk+ zk) − fk+1 fk (xk−1+ zk−1) pk/M ≤ sup k∈N ffk k+1 xk− fk+1 fk xk−1 pk/M + sup k∈N ffk k+1 zk− fk+1 fk zk−1 pk/M (5) and for any α ∈ R (see [2]),
|α|pk
It is clear that g(θ) = 0 and g(x) = g(−x) for all x ∈ c0( bF , p). Again the
inequalities (5) and (6) yield the subadditivity of g and g(αx) ≤ max{1, |α|}g(x). Let {xn} be any sequence of the points xn ∈ c
0( bF , p) such that g(xn−x) → 0
and (αn) also be any sequence of scalars such that αn → α. Then, since the
inequality
g(xn) ≤ g(x) + g(xn− x)
holds by the subadditivity of g, {g(xn)} is bounded and we thus have
g(αnxn− αx) = sup k∈N ffk k+1 (αnxnk− αxk) − fk+1 fk (αnxnk−1− αxk−1) pk/M ≤ |αn− α| g(xn) + |α| g(xn− x),
which tends to zero as n → ∞. That is to say that the scalar multiplication is continuous. Hence, g is a paranorm on the space c0( bF , p).
It remains to prove the completeness of the space c0( bF , p). Let {xi} be
any Cauchy sequence in the space c0( bF , p), where xi = {x(i)0 , x (i)
1 , ...}. Then,
for a given ε > 0 there exists a positive integer n0(ε) such that
g(xi− xj) < ε 2
for all i, j ≥ n0(ε). We obtain by using definition of g for each fixed k ∈ N
that |{ bF xi}k− { bF xj}k|pk/M ≤ sup k∈N|{ b F xi}k− { bF xj}k|pk/M < ε 2 (7)
for every i, j ≥ n0(ε), which leads us to the fact that {( bF x0)k, ( bF x1)k, ...} is a
Cauchy sequence of real numbers for every fixed k ∈ N. Since R is complete, it converges, say
{ bF xi}k→ { bF x}k
as i → ∞. Using these infinitely many limits ( bF x)0, ( bF x)1, ..., we define the
sequence {( bF x)0, ( bF x)1, ...}. We have from (7) with j → ∞ that
|{ bF xi}k− { bF x}k|p
k/M
for every fixed k ∈ N. Since xi = {x(i) k } ∈ c0( bF , p), |{ bF xi}k|p k/M < ε 2 for all k ∈ N. Therefore, we obtain (8) that
|{ bF x}k|p k/M ≤ |{ bF x}k− { bF xi}k|p k/M + |{ bF xi} k|p k/M < ε (i ≥ n0(ε)).
This shows that the sequence { bF x} belongs to the space c0(p). Since {xi}
was an arbitrary Cauchy sequence, the space c0( bF , p) is complete and this
concludes the proof.
Therefore, one can easily check that the absolute property does not hold on the spaces c0( bF , p), c( bF , p) , ℓ∞( bF , p) and ℓ( bF , p) that is h(x) 6= h(|x|)
for at least one sequence in those spaces, and this says that c0( bF , p), c( bF , p)
,ℓ∞( bF , p) and ℓ( bF , p) are the sequence spaces of non-absolute type; where
|x| = (|xk|).
Theorem 2. The sequence spaces c0( bF , p), c( bF , p) ,ℓ∞( bF , p) and ℓ( bF , p) are
linearly isomorphic to the spaces c0(p), c(p), ℓ∞(p) and ℓ(p), respectively, where
0 < pk≤ H < ∞.
Proof. We establish this for the space ℓ∞( bF , p). To prove the theorem, we should show the existence of a linear bijection between the spaces ℓ∞( bF , p)
and ℓ∞(p) for 0 < pk ≤ H < ∞. With the notation of (4), define the
transformations T from ℓ∞( bF , p) to ℓ∞(p) by x 7→ y = T x. The linearity of
T is trivial. Further, it is obvious that x = θ whenever T x = θ and hence T is injective.
Let y = (yk) ∈ ℓ∞(p) and define the sequence x = (xk) by
xk = k X j=0 f2 k+1 fjfj+1 yj; (k ∈ N).
Then, we get that g(x) = sup k∈N ffk k+1 xk− fk+1 fk xk−1 pk/M = sup k∈N ffk k+1 k X j=0 fk+12 fjfj+1 yj − fk+1 fk k−1 X j=0 f2 k fjfj+1 yj pk/M = sup k∈N|yk| pk/M = h1(y) < ∞.
Thus, we deduce that x ∈ ℓ∞( bF , p) and consequently T is surjective and is
paranorm preserving. Hence, T is a linear bijection and this says us that the spaces ℓ∞( bF , p) and ℓ∞(p) are linearly isomorphic, as desired.
We shall quote some lemmas which are needed in proving related to the duals our theorems.
Lemma 1. [21, Theorem 5.1.1 with qn= 1] A ∈ (c0(p) : ℓ(q)) if and only if
sup K∈F X n X k∈K ankB−1/pk < ∞, (∃B ∈ N2). (9) Lemma 2. [21, Theorem 5.1.9 with qn= 1] A ∈ (c0(p) : c(q)) if and only if
sup n∈N X k |ank|B−1/pk < ∞ (∃B ∈ N2), (10) ∃(αk) ⊂ R ∋ lim n→∞|ank− αk| = 0 for all k ∈ N, (11) ∃(αk) ⊂ R ∋ sup n∈N X k |ank− αk|B −1/pk < ∞. (∃B ∈ N2) (12)
Lemma 3. [21, Theorem 5.1.13 with qn = 1] A ∈ (c0(p) : ℓ∞(q)) if and only
if sup n∈N X k |ank|B−1/pk < ∞. (∃B ∈ N2) (13)
Lemma 4. [21, Theorem 5.1.0 with qn = 1](i) Let 1 < pk≤ H < ∞ for all
k ∈ N. Then, A ∈ (ℓ(p) : ℓ1) if and only if there exists an integer B > 1 such
that sup K∈F X k X n∈K ankB−1 p′k < ∞. (14)
(ii) Let 0 < pk ≤ 1 for all k ∈ N. Then, A ∈ (ℓ(p) : ℓ1) if and only if
sup K∈F sup k∈N X n∈K ank pk < ∞. (15)
Lemma 5. [21, Theorem 1 (i)-(ii)] (i) Let 1 < pk ≤ H < ∞ for all k ∈ N.
Then, A ∈ (ℓ(p) : ℓ∞) if and only if there exists an integer B > 1 such that
sup n∈N X k |ankB−1|p ′ k < ∞. (16)
(ii) Let 0 < pk ≤ 1 for all k ∈ N. Then, A ∈ (ℓ(p) : ℓ∞) if and only if
sup
n,k∈N|ank| pk
< ∞. (17)
Lemma 6. [21, Corollary for Theorem 1] Let 0 < pk ≤ H < ∞ for all
k ∈ N. Then, A ∈ (ℓ(p) : c) if and only if (16), (17) hold, and lim
n→∞ank = βk, (k ∈ N) (18)
also holds.
Theorem 3. Let K∗
= {k ∈ N : 0 ≤ k ≤ n} ∩ K for K ∈ F and B ∈
N2. Define the sets bF1(p), bF2(p), bF3(p), bF4(p), bF5(p), bF6(p), bF7(p) and bF8(p)
as follows: b F1(p) = [ B>1 n a = (ak) ∈ ω : sup K∈F X n X k∈K∗ f2 n+1 fkfk+1 anB−1/pk < ∞o b F2(p) = n a = (ak) ∈ ω : X n n X k=0 f2 n+1 fkfk+1 an < ∞o b F3(p) = [ B>1 n a = (ak) ∈ ω : sup n∈N n X k=0 n X j=k f2 j+1 fkfk+1 aj B−1/pk < ∞o b F4(p) = n a = (ak) ∈ ω : ∞ X j=k f2 j+1 fkfk+1 aj < ∞ for all k ∈ No b F5(p) = [ B>1 n a = (ak) ∈ ω : ∃(αk) ⊂ R ∋ sup n∈N n X k=0 n X j=k f2 j+1 fkfk+1 aj − αk B−1/pk < ∞o b F6(p) = n a = (ak) ∈ ω : ∃α ∈ R ∋ lim n→∞ n X k=0 n X j=k f2 j+1 fkfk+1 aj − α = 0o b F7(p) = n a = (ak) ∈ ω : sup n∈N n X k=0 n X j=k f2 j+1 fkfk+1 aj < ∞o
Then, (i) {c0( bF , p)}α = bF1(p) (ii) {c( bF , p)}α = bF1(p) ∩ bF2(p) (iii) {c0( bF , p)}β = bF3(p) ∩ bF4(p) ∩ bF5(p) (iv) {c( bF , p)}β = {c 0( bF , p)}β ∩ bF6(p) (v) {c0( bF , p)}γ = bF3(p) (vi) {c( bF , p)}γ = bF3(p) ∩ bF7(p)
Proof. We give the proof for the space c0( bF , p). Let us take any a = (an) ∈ ω and define the matrix C = (cnk) via the sequence a = (an) by
cnk = f2 n+1 fkfk+1 an, 0 ≤ k ≤ n, 0, k > n,
where n, k ∈ N. Bearing in mind (4) we immediately derive that anxn= n X k=0 f2 n+1 fkfk+1 anyk= (Cy)n; (n ∈ N). (19)
We therefore observe by (19) that ax = (anxn) ∈ ℓ1 whenever x ∈ c0( bF , p)
if and only if Cy ∈ ℓ1 whenever y ∈ c0(p). This means that a = (an) ∈
{c0( bF , p)}α whenever x = (xn) ∈ c0( bF , p) if and only if C ∈ (c0(p) : ℓ1).
Then, we derive by Lemma 1 that
{c0( bF , p)}α = bF1(p).
Consider the equation for n ∈ N,
n X k=0 akxk = n X k=0 ak Xn j=0 f2 k+1 fjfj+1 yj = n X k=0 Xn j=k f2 j+1 fkfk+1 aj yk = (Dy)n (20) where D = (dnk) is defined by dnk = n X j=k f2 j+1 fkfk+1 aj, 0 ≤ k ≤ n, 0, k > n,
where n, k ∈ N. Thus, we deduce from Lemma 2 with (20) that ax = (akxk) ∈
cs whenever x = (xk) ∈ c0( bF , p) if and only if Dy ∈ c whenever y = (yk) ∈
c0(p). This means that a = (an) ∈ {c0( bF , p)}β whenever x = (xn) ∈ c0( bF , p)
if and only if D ∈ (c0(p) : c). Therefore we derive from Lemma 2 that
{c0( bF , p)}β = bF3(p) ∩ bF4(p) ∩ bF5(p).
As this, we deduce from Lemma 3 with (20) that ax = (akxk) ∈ bs
whenever x = (xk) ∈ c0( bF , p) if and only if Dy ∈ ℓ∞ whenever y = (yk) ∈
c0(p). This means that a = (an) ∈ {c0( bF , p)}γ whenever x = (xn) ∈ c0( bF , p)
if and only if D ∈ (c0(p) : ℓ∞). Therefore we obtain Lemma 3 that
{c0( bF , p)}γ = bF3(p)
and this completes the proof. Theorem 4. Let K∗
= {k ∈ N : 0 ≤ k ≤ n} ∩ K for K ∈ F and B ∈ N2.
Define the sets bF8(p), bF9(p), bF10(p) and bF11(p) as follows:
b F8(p) = \ B>1 n a = (ak) ∈ ω : sup K∈F X n X k∈K∗ n X j=k fj+12 fkfk+1 ajB1/pk < ∞o b F9(p) = \ B>1 n a = (ak) ∈ ω : sup n∈N n X k=0 n X j=k f2 j+1 fkfk+1 aj B1/pk < ∞o b F10(p) = \ B>1 n a = (ak) ∈ ω : ∃(αk) ⊂ R ∋ lim n→∞ n X k=0 n X j=k f2 j+1 fkfk+1 aj − αk B1/pk = 0o b F11(p) = \ B>1 n a = (ak) ∈ ω : sup n∈N n X k=0 n X j=k f2 j+1 fkfk+1 aj B1/pk < ∞o Then, (i) {ℓ∞( bF , p)}α= bF8(p) (ii) {ℓ∞( bF , p)}β = bF9(p) ∩ bF10(p) (iii) {ℓ∞( bF , p)}γ = bF11(p).
Proof. This may be obtained in the similar way, as mentioned in the proof of Theorem 3 with Lemmas 4(i), 5(i), 6 instead of Lemmas 1-3. So, we omit the details.
Theorem 5. Let K∗
= {k ∈ N : 0 ≤ k ≤ n} ∩ K for K ∈ F and B ∈ N2.
Define the sets bF12(p), bF13(p), bF14(p), bF15(p) and bF16(p) as follows:
b F12(p) = n a = (ak) ∈ ω : sup K∈F sup k∈N X n∈K∗ n X j=k f2 j+1 fkfk+1 aj pk < ∞o b F13(p) = [ B>1 n a = (ak) ∈ ω : sup K∈F X k X n∈K n X j=k f2 j+1 fkfk+1 ajB−1 p′k < ∞o b F14(p) = [ B>1 n a = (ak) ∈ ω : sup n∈N n X k=0 n X j=k f2 j+1 fkfk+1 ajB−1 p′k < ∞o b F15(p) = n a = (ak) ∈ ω : sup n,k∈N n X j=k f2 j+1 fkfk+1 aj pk < ∞o b F16(p) = n a = (ak) ∈ ω : lim n→∞ n X j=k f2 j+1 fkfk+1 aj exists o Then, (i) {ℓ( bF , p)}α = ( b F12(p), 0 < pk≤ 1 b F13(p), 1 < pk≤ H < ∞ (ii) {ℓ( bF , p)}γ = ( b F15(p), 0 < pk≤ 1 b F14(p), 1 < pk≤ H < ∞.
(iii) Let 0 < pk ≤ H < ∞. Then,
{ℓ( bF , p)}β = bF
14(p) ∩ bF15(p) ∩ bF16(p).
Proof. This may be obtained in the similar way, as mentioned in the proof of Theorem 3 with Lemmas 4(ii), 5(ii), 6 instead of Lemmas 1-3. So, we omit the details.
Now, we may give the sequence of the points of the spaces c0( bF , p), ℓ( bF , p)
and c( bF , p) which forms a Schauder basis for those spaces. Because of the isomorphism T , defined in the proof of Theorem 2, between the sequence spaces c0( bF , p) and c0(p), ℓ( bF , p) and ℓ(p), c( bF , p) and c(p) is onto, the
inverse image of the basis of the spaces c0(p), ℓ(p) and c(p) is the basis for
our new spaces c0( bF , p), ℓ( bF , p) and c( bF , p), respectively. Therefore, we have:
Theorem 6. Let µk = ( bF x)k for all k ∈ N. We define the sequence b(k) =
{b(k)n }n∈N for every fixed k ∈ N by
b(k)n = f2 n+1 fkfk+1, n ≥ k, 0, n < k. Then,
(a) The sequence {b(k)}
k∈N is a basis for the space c0( bF , p) and any x ∈
c0( bF , p) has a unique representation in the form
x =X
k
µkb(k).
(b) The sequence {b(k)}
k∈N is a basis for the space ℓ( bF , p) and any x ∈ ℓ( bF , p)
has a unique representation in the form
x =X
k
µkb(k).
(c) The set {z, b(k)} is a basis for the space c( bF , p) and any x ∈ c( bF , p) has
a unique representation in the form
x = lz +X
k
(µk− l)b(k)
where l = limk→∞( bF x)k and z = (zk) with
zk = k X j=0 f2 k+1 fjfj+1 .
3. Some Matrix Mappings on the Sequence Spaces c0( bF , p), c( bF , p)
, ℓ∞( bF , p) and ℓ( bF , p)
In this section, we characterize some matrix mappings on the spaces c0( bF , p), c( bF , p), ℓ∞( bF , p) and ℓ( bF , p). Firstly, we may give the following
the-orem which is useful for deriving the characterization of the certain matrix classes.
Theorem 7. [22, Theorem 4.1] Let λ be an FK-space, U be a triangle, V be
its inverse and µ be arbitrary subset of ω. Then we have A ∈ (λU : µ) if and
only if E(n)= (e(n)mk) ∈ (λ : c) for all n ∈ N (21) and E = (enk) ∈ (λ : µ) (22) where e(n)mk = m X j=k anjvjk, 0 ≤ k ≤ m, 0, k > m, and enk = ∞ X j=k anjvjk for all k, m, n ∈ N.
Now, we may quote our theorems on the characterization of some matrix classes concerning with the sequence spaces c0( bF , p), c( bF , p) and ℓ∞( bF , p).
The necessary and sufficient conditions characterizing the matrix mappings between the sequence spaces of Maddox are determined by Grosse-Erdmann [21]. Let N and K denote the finite subset of N, L and M also denote the natural numbers. Prior to giving the theorems, let us suppose that (qn)
is a non-decreasing bounded sequence of positive numbers and consider the following conditions: lim m→∞ m X j=k f2 j+1 fkfk+1 anj = enk, (23) ∀L, X k |enk|L1/pk < ∞, (24) ∃(αk) ⊂ R ∋ lim m→∞ m X j=k f2 j+1 fkfk+1 anj− αk = 0 for all k ∈ N, (25) ∃M, sup m∈N m X k=0 m X j=k f2 j+1 fkfk+1 anj M−1/pk < ∞, (26) ∀L, ∃M, sup m∈N m X k=0 m X j=k f2 j+1 fkfk+1 anj L1/qn M−1/pk < ∞, (27)
lim m→∞ X k m X j=k f2 j+1 fkfk+1 anj − α = 0, (28) ∀L, sup n∈N X k |enk|L1/pk < ∞, (29) lim n→∞enk = αk for all k ∈ N, (30) ∀L, lim n→∞ X k |enk|L1/pk < ∞, (31) ∀L, lim n→∞ X k |enk|L1/pk = 0, (32) ∃M, sup n∈N X k∈K |enk|M−1/pk qn < ∞, (33) lim n→∞|enk| qn = 0, for all k ∈ N, (34) ∀L, ∃M, sup n∈N X k |enk|L1/qnM −1/pk < ∞, (35) lim n→∞|enk− αk| qn = 0, for all k ∈ N, (36) ∃M, sup n∈N X k |enk|M−1/pk < ∞, (37) ∀L, ∃M, sup n∈N X k |enk− αk|L1/qnM−1/pk < ∞, (38) sup n∈N X k enk qn < ∞, (39) lim n→∞ X k enk qn = 0, (40) lim n→∞ X k enk− α qn = 0, (41)
Theorem 8. (i) A ∈ (ℓ∞( bF , p) : ℓ∞) if and only if (23),(24) and (29) hold.
(ii) A ∈ (ℓ∞( bF , p) : c) if and only if (23),(24), (30) and (31) hold.
Theorem 9. (i) A ∈ (c0( bF , p) : ℓ∞(q)) if and only if (25), (26), (27) and
(33) hold.
(ii) A ∈ (c0( bF , p) : c0(q)) if and only if (25), (26), (27), (34) and (35)
hold.
(iii) A ∈ (c0( bF , p) : c(q)) if and only if (25), (26), (27), (36), (37) and
(38) hold.
Theorem 10. (i) A ∈ (c( bF , p) : ℓ∞(q)) if and only if (25), (26), (27), (28),
(33) and (39) hold.
(ii) A ∈ (c( bF , p) : c0(q)) if and only if (25), (26), (27), (28), (34), (35)
and (40) hold.
(iii) A ∈ (c( bF , p) : c(q)) if and only if (25), (26), (27), (28), (36), (37),
(38) and (41) hold.
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